Sets, Relations & Functions PYQs Set 16 (PDF)

Set 16 of 25 · Questions 451-480 of 734

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 16 (Q451-480)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
451. $A function y = f (x) with f (- 1) = - 249 has no maximum and has only one minimum at x = 5 with f (5) = 75 . Which one of the following is true?$ (1) $At some point in (- 1, 5), f (x) is discontinuous$ (2) $The minimum value cannot be 75 since f (- 1) < f (5) = 75$ (3) $f (x) is discontinuous at every point of R$ (4) $f (x) is continuous on R$ 452. $lim x \to 0 \frac{x ^{2} + 100 - 10}{x ^{2}} =$ (1) $0$ (2) $0.1$ (3) $0.05$ (4) $-0.05$ 453. $If sec^{- 1} (\frac{1 + x}{1 - y}) = a, then \frac{dy}{dx} is$ (1) $\frac{y - 1}{x + 1}$ (2) $\frac{y + 1}{x - 1}$ (3) $\frac{x - 1}{y - 1}$ (4) $\frac{x - 1}{y + 1}$ 454. $Let f : R \to R and g : R \to R be defined by f (x) = 2 x + 1 and g (x) = x^{2} - 2 deternine (gof) (x) is equal to$ (1) $2 x^{2} - 3$ (2) $4 x^{2} + 4 x - 1$ (3) $4 x^{2} + 4 x + 1$ (4) $2 x^{2} - 4$ 455. $The money invested in a company is compounded continuously. If ₹ 200 invested today becomes ₹ 400 in 6 years, then at the end of 33 years it will become ₹$ (1) $16002$ (2) $32002$ (3) $128002$ (4) $64002$ 456. $To set of all real values of x for which f (x) = log_{2} (2^{x} - 2) + 1 - x is also real is$ (1) $R$ (2) $(1, \infty)$ (3) $(- \infty, 1]$ (4) $ϕ$ 457. $If A is the domain and B is the range of the function f (x) = (3 x - 1, x > 1 x^{2} + 1, x \leq 1), then A - B =$ (1) $(1, \infty)$ (2) $(- \infty, 1)$ (3) $R - (- 1, 1)$ (4) $(- 1, 1)$ 458. $If [x] represents the greatest integer function and f (x) = x - [x] - cos x, then f^{’} (\frac{π}{2}) =$ (1) $0$ (2) $2$ (3) $1$ (4) $does not exist$ 459. $If y = \frac{e ^{s i n x} + s i n h ^{3} x}{c o s h x - t a n x}, then y^{'} (0) =$ (1) $0$ (2) $1$ (3) $-1$ (4) $2$ 460. $If the domain of the function sin^{- 1} (\frac{3 x - 22}{2 x - 19}) + log_{e} (\frac{3 x ^{2} - 8 x + 5}{x ^{2} - 3 x - 10}) is (α, β], then 3 α + 10 β is equal to:$ (1) $100$ (2) $95$ (3) $97$ (4) $98$ 461. $If x = a sec^{2} θ, y = a tan^{2} θ, then \frac{d ^{2} y}{d x ^{2}} =$ (1) $2 a$ (2) $0$ (3) $1$ (4) $4$ 462. $If 2^{x} + 2^{y} = 2^{x + y}, then \frac{dy}{dx} is$ (1) $- 2^{y - x}$ (2) $2^{x - y}$ (3) $\frac{2 ^{y - 1}}{2 ^{x - 1}}$ (4) $2^{y - x}$ 463. $Consider the relations R_{1} and R_{2} defined as a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1 for all a, b, \in R and (a, b) R_{2} (c, d) \Leftrightarrow a + d = b + c for all (a, b), (c, d) \in N \times N . Then$ (1) $Only R_{1} is an equivalence relation$ (2) $Only R_{2} is an equivalence relation$ (3) $R_{1} and R_{2} both are equivalence relations$ (4) $Neither R_{1} nor R_{2} is an equivalence relation$ 464. $Let a, b \in R and f : R \to R be defined by f(x) = a cos (\| x^{3} -x\|) + b\|x\| sin(\| x^{3} +x\|). Then f is$ (1) $differentiable at x = 0 if a = 0 and b = 1$ (2) $differentiable at x = 1 if a = 1 and b = 0$ (3) $NOT differentiable at x = 0 if a = 1 and b = 0$ (4) $NOT differentiable at x = 1 if a = 1 and b = 1$ 465. $Let f (x) = \frac{1}{7 - sin 5 x} be a function defined on R . Then the range of the function f (x) is equal to ;$ (1) $(\frac{1}{7}, \frac{1}{6})$ (2) $(\frac{1}{8}, \frac{1}{5})$ (3) $(\frac{1}{7}, \frac{1}{5})$ (4) $(\frac{1}{8}, \frac{1}{6})$ 466. $If tan^{- 1} (x^{2} + y^{2}) = α then \frac{dy}{dx} is equal to$ (1) $\frac{- x}{y}$ (2) $xy$ (3) $\frac{x}{y}$ (4) $- xy$ 467. $Consider the function f : (\frac{1}{2}, 1) \to R defined by f (x) = 42 x^{3} - 32 x - 1 . Consider the statements (I) The curve y = f (x) intersects the x -axis exactly at one point (II) The curve y = f (x) intersects the x -axis at x = cos \frac{π}{12} Then$ (1) $Only (II) is correct$ (2) $Both (I) and (II) are incorrect$ (3) $Only (I) is correct$ (4) $Both (I) and (II) are correct$ 468. $lim x \to a \frac{a + 2 x - 3 a}{x - a} =$ (1) $- \frac{5}{3}$ (2) $- \frac{1}{3}$ (3) $\frac{1}{3}$ (4) $\frac{2}{3}$ 469. $Let f : R - {2, 6} \to R be real valued function defined as f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} - 8 x + 12} . Then range of f is$ (1) $(- \infty, - \frac{21}{4}) \cup [0, \infty)$ (2) $(- \infty, - \frac{21}{4}) \cup (0, \infty)$ (3) $(- \infty, - \frac{21}{4}) \cup (\frac{21}{4}, \infty)$ (4) $(- \infty, - \frac{21}{4}) \cup (1, \infty)$ 470. $Let the relation R on the set M = 1, 2, 3, \dots, 16 be given by R = (x, y) : 4y=5x-3, x, y \in M . Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to$ (1) $4$ (2) $1$ (3) $2$ (4) $3$ 471. $For A \neq = 0, x < 0, lim_{n \to \infty} \frac{sin x - e ^{nx}}{1 + A e ^{nx}} =$ (1) $\frac{1}{A}$ (2) $sin x$ (3) $- \frac{1}{A}$ (4) $- sin x$ 472. $lim_{x \to 0} \frac{6 ^{x} - 3 ^{x} - 2 ^{x} + 1}{x ^{2}} is equal to$ (1) $(log_{e} 2) log_{e} 3$ (2) $log_{e} 5$ (3) $log_{e} 6$ (4) $0$ 473. $If the domain of the function f (x) = sin^{- 1} (\frac{5 - x}{3 + 2 x}) + \frac{1}{log _{e} ( 10 - x )} is (- \infty, α] \cup [β, γ) - {δ}, then 6 (α + β + γ + δ) is equal to$ (1) $67$ (2) $66$ (3) $70$ (4) $68$ 474. $If y = 2 x + cos^{2} (2 x + \frac{π}{4}), then \frac{dy}{dx} at x = \frac{π}{4} .$ (1) $\frac{2 2}{π + 1}$ (2) $2 π + 1$ (3) $\frac{2}{π + 1}$ (4) $\frac{2}{π + 1}$ 475. $Let R be a relation on N \times N defined by (a, b) R (c, d) if and only if ad (b - c) = bc (a - d) . Then R is$ (1) $Symmetric but neither reflexive nor Transitive$ (2) $Trasitive but neither reflexive nor symmetric$ (3) $Reflexive and symmetric but not transitive$ (4) $symmetric and transitive but not reflexive$ 476. $If the domain of the function f (x) = \frac{x ^{2} - 25}{( 4 - x ^{2} )} + log_{10} (x^{2} + 2 x - 15) is (- \infty, α) U [β, \infty), then α^{2} + β^{3} is equal to :$ (1) $140$ (2) $175$ (3) $150$ (4) $125$ 477. $lim_{x \to 0} \frac{1 - cos x}{x ^{2}} is$ (1) $3$ (2) $\frac{1}{3}$ (3) $2$ (4) $\frac{1}{2}$ 478. $For constant a, \frac{d}{dx} (x^{x} + x^{a} + a^{x} + a^{a}) is$ (1) $x^{x} (1 + log x) + a x^{a - 1}$ (2) $x^{x} (1 + log x) + a x^{a - 1} + a^{x} log a$ (3) $x^{x} (1 + log x) + a^{a} (1 + log x)$ (4) $x^{x} (1 + log x) + a^{a} (1 + log a) + a x^{a - 1}$ 479. $Let A = 1, 2, 3, 4 and R = (1, 2), (2, 3), (1, 4) be a relation on A . Let S be the equivalence relation on A such that R \subset S and the number of elements in S is n . Then, the minimum value of n is______$ 480. $Let A = 1, 2, 3, 4, 5, 6, 7 and B = 3, 6, 7, 9 . Then the number of elements in the set {C \subseteq A : C \cap B \neq = φ} is____$